Kaprekar Number 6174

Few days back I posted an article based on the interesting properties of 153. Lot of people got very excited to know about similar other numbers with such interesting properties. Today I will be discussing about another such interesting number: 6174

Kaprekar’s Constant

6174 is known as Kaprekar’s constant named after the Indian recreational mathematician D. R. Kaprekar. I’ve written about D.R.Kaprekar and contribution of other Indian mathematicians. 6174 has got a very interesting property. To know what that mysterious property is take any four-digit number. Arrange the digits in ascending and then in descending order to get two four-digit numbers. Then subtract the bigger number from the smaller number. If we keep on repeating this process we will end up in 6174. This process is called Kaprekar’sroutine. All the numbers will yield 6174 in 7 or less than 7 iterations.

Example

Let’s randomly choose any number, say 4518:

Now, arranging the digits in ascending and then in descending order to get two four-digit numbers.

8541-1458 = 7083

8730-0378 = 8352

8532-2358 = 6174

Hence we get 6174 in 3 iterations.

4651 reaches 6174 after 7 iterations

6541-1456 = 5085

8550-558 = 7992

9972-2799 = 7173

7731-1377 = 6354

6543-3456 = 3087

8730-378 = 8352

8532-2358 = 6174

Try it for any 4-digit number yourself and see if it works.

Questions

For a specific set of numbers Kaprekar’s routine will not work. Can you tell me what numbers will those be?

If you follow Kaprekar routine with any 3 digit number it will also result in one specific number. Can you find out that 3 digit equivalent constant?

The result of each iteration of Kaprekar’s routine is a multiple of 9. Can you explain why?

Hint: you have seen the application of similar mathematical logic in the earlier post – mind reading trick.

1) The answer to 1st question is that all the numbers with repeating digits are an exception like 3333, 4444, 5555 etc.

2) The number should be 297

3) This is because whenever one subtracts a number from its reverse number it is always divisible by 9

a four digit number can be written as (1000x + 100y + 10z + w) – (1000w + 100z + 10y + x) = 999x + 90y – 90z – 999x which will always be divisible by 9. The same rule applies to the 3 digit number above.